摘要
Let <em>G</em> be a group. <em>G</em> is right-orderable provided it admits a total order ≤ satisfying <em>hg</em><sub>1</sub> <span style="white-space:normal;">≤ <span style="white-space:normal;"><em>hg</em><sub>2 </sub></span></span>whenever <em style="white-space:normal;">g</em><sub style="white-space:normal;">1</sub><span style="white-space:normal;"> </span><span style="white-space:normal;">≤ <i>g</i><sub>2</sub></span>. <em>G</em> is orderable provided it admits a total order ≤ satisfying both: <em style="white-space:normal;">hg</em><sub style="white-space:normal;">1</sub><span style="white-space:normal;"> </span><span style="white-space:normal;">≤ <em>hg</em><sub>2</sub></span> whenever <span style="white-space:nowrap;"><em>g</em><sub>1</sub> ≤ <em>g</em><sub>2</sub></span> and <em style="white-space:normal;">g</em><sub style="white-space:normal;">1</sub><em style="white-space:normal;">h</em><span style="white-space:normal;"> ≤ </span><em style="white-space:normal;">g</em><sub style="white-space:normal;">2</sub><em style="white-space:normal;">h</em> whenever <em>g</em><sub>1</sub> ≤ <em>g</em><sub>2</sub>. A classical result shows that free groups are orderable. In this paper, we prove that left-orderable groups and orderable groups are quasivarieties of groups both with undecidable theory. For orderable groups, we find an explicit set of universal axioms.
Let <em>G</em> be a group. <em>G</em> is right-orderable provided it admits a total order ≤ satisfying <em>hg</em><sub>1</sub> <span style="white-space:normal;">≤ <span style="white-space:normal;"><em>hg</em><sub>2 </sub></span></span>whenever <em style="white-space:normal;">g</em><sub style="white-space:normal;">1</sub><span style="white-space:normal;"> </span><span style="white-space:normal;">≤ <i>g</i><sub>2</sub></span>. <em>G</em> is orderable provided it admits a total order ≤ satisfying both: <em style="white-space:normal;">hg</em><sub style="white-space:normal;">1</sub><span style="white-space:normal;"> </span><span style="white-space:normal;">≤ <em>hg</em><sub>2</sub></span> whenever <span style="white-space:nowrap;"><em>g</em><sub>1</sub> ≤ <em>g</em><sub>2</sub></span> and <em style="white-space:normal;">g</em><sub style="white-space:normal;">1</sub><em style="white-space:normal;">h</em><span style="white-space:normal;"> ≤ </span><em style="white-space:normal;">g</em><sub style="white-space:normal;">2</sub><em style="white-space:normal;">h</em> whenever <em>g</em><sub>1</sub> ≤ <em>g</em><sub>2</sub>. A classical result shows that free groups are orderable. In this paper, we prove that left-orderable groups and orderable groups are quasivarieties of groups both with undecidable theory. For orderable groups, we find an explicit set of universal axioms.
作者
Benjamin Fine
Anthony Gaglione
Gerhard Rosenberger
Dennis Spellman
Benjamin Fine;Anthony Gaglione;Gerhard Rosenberger;Dennis Spellman(Department of Mathematics, Fairfield University, Fairfield, Connecticut, USA;Department of Mathematics, United States Naval Academy, Annapolis, Maryland, USA;Fachbereich Mathematik, University of Hamburg, Bundestrasse, Hamburg, Germany;Department of Statistics, Temple University, Philadelphia, Pennsylvania, USA)
